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  • Cited by 6
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bernicot, Frédéric Coulhon, Thierry and Frey, Dorothee 2016. Gaussian heat kernel bounds through elliptic Moser iteration. Journal de Mathématiques Pures et Appliquées,

    Chen, Li Coulhon, Thierry Feneuil, Joseph and Russ, Emmanuel 2016. Riesz Transform for $$1\le p \le 2$$ 1 ≤ p ≤ 2 Without Gaussian Heat Kernel Bound. The Journal of Geometric Analysis,

    Kaynak, Burak Tevfik and Turgut, O. Teoman 2012. A Klein–Gordon particle captured by embedded curves. Annals of Physics, Vol. 327, Issue. 11, p. 2605.

    Robinson, Derek W. and Sikora, Adam 2008. Analysis of degenerate elliptic operators of Grušin type. Mathematische Zeitschrift, Vol. 260, Issue. 3, p. 475.

    Telcs, András 2007. Lower bounds for transition probabilities on graphs. Stochastic Processes and their Applications, Vol. 117, Issue. 8, p. 1121.

    Barlow, Martin T. Coulhoun, Thierry and Kumagai, Takashi 2005. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Communications on Pure and Applied Mathematics, Vol. 58, Issue. 12, p. 1642.

  • Journal of the London Mathematical Society, Volume 68, Issue 3
  • December 2003, pp. 795-816


  • DOI:
  • Published online: 17 November 2003

On a manifold with polynomial volume growth satisfying Gaussian upper bounds of the heat kernel, a simple characterization of the matching lower bounds is given in terms of a certain Sobolev inequality. The method also works in the case of so-called sub-Gaussian or sub-diffusive heat kernels estimates, which are typical of fractals. Together with previously known results, this yields a new characterization of the full upper and lower Gaussian or sub-Gaussian heat kernel estimates.

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Research partially supported by the European Commission (IHP Network ‘Harmonic analysis and related problems’ 2002–2006, contract HPRN-CT-2001-00273-HARP).
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Journal of the London Mathematical Society
  • ISSN: 0024-6107
  • EISSN: 1469-7750
  • URL: /core/journals/journal-of-the-london-mathematical-society
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